Solve $\int \frac{\sin(x)+\cos(x)}{e^x}~dx$

Question

How should I calculate this integral$$\int\frac{\sin(x)+\cos(x)}{e^x}dx~~~~?$$ I don't know what is the first step, so I tried symbolab calculator but this cannot solve. Can someone help me to solve this? Thank you in advance.

Answer 1

Your integral can be re-written as $$I=\sqrt{2}\int e^{-x} \sin (x+\pi/4)~ dx.$$ You may do it by parts, two times. Second time $I$ will re-appear, take it LHS to get the final result.

Answer 2

$$\dfrac{d(e^{-x}(a\cos x+b\sin x)}{dx}$$

$$=e^{-x}(-a\sin x+b\cos x-a\cos x-b\sin x)$$

Compare the coefficients of $e^{-x}\cos x,e^{-x}\sin x$ with $$e^{-x}(\cos x+\sin x)$$ to find

$b-a=1,-a-b=1,a=?,b=?$

Answer 3

Note the integral fornula: $$\int e^{-x}(f(x)-f'(x)) dx=-e^{-x} f(x).$$ So here $$\int e^{-x} (\cos x+\sin x) dx=-e^{-x} \cos x +C$$